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It is well known that some dispersive non--linear equations admit traveling wave solutions $$ u(t,x)=u_0(x-ct)\in L^2_x\,, \qquad (t,x)\in \mathbb{R}\times \mathbb{R}\,\text{ or }\, \mathbb{R}\times\mathbb{T}\,, $$ where $u_0$ is the profile and $c$ is a real constant. Sometimes these traveling waves can be obtained as ground states (minimizers) of the energy, mass... functionals of the equation, leading to say that there exists an $L^2$--threshold below (or above) which the existence of traveling waves cannot occur.

My question : Do there exist nonlinear dispersive PDEs, which have traveling waves with large and small $L^2$ norms ?

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  • $\begingroup$ your interest is in solitonic waves, right, not general traveling waves? $\endgroup$
    – Carlo Beenakker
    Commented Feb 12, 2023 at 14:22
  • $\begingroup$ @CarloBeenakker I was first thinking about traveling waves. But, since you mentioned solitonic waves, I also have the same question for solitons... $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 9:43

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This addresses the case of a traveling wave (not a solitonic wave).

If I consider the one-dimensional nonlinear Schrödinger equation $$i\partial_t\Psi+\partial_x^2\Psi+\Psi(f(|\Psi|^2)=0,$$ a travelling wave $\Psi(t,x)=U(x-ct)$ which tends to $|U(x)|\rightarrow r_0>0$ for $x\rightarrow\pm\infty$ exists if the velocity $c$ is not too large, $$c<\sqrt{2r_0^2|f'(r_0^2)|}\equiv c^\ast(r_0).$$ So the question whether $U$ can have arbitrarily large norm in a finite system (with periodic boundary conditions) becomes the question whether $c^\ast(r_0)$ can become arbitrarily large with increasing $r_0$, which is possible, for example, if $f(x)=1-x$ (Gross-Pitaevskii equation).

See Traveling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension one.

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  • $\begingroup$ My following question might be silly: Suppose that $c^*(r_0)$ is bounded. I don't understand why if $c<c^*(r_0)$ then this implies the $L^2$-norm of $U$ cannot be arbitrarily large. $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 11:26
  • $\begingroup$ perhaps I misunderstood your question, but first of all, for a travelling wave we need a finite system for a finite norm (since by definition, $u_0(x)$ does not go to zero for large $x$). We would typically place the system on a torus, and impose periodic boundary conditions on a length $L$; then we ask whether, for a fixed $L$, the norm of $u_0$ can become arbitrarily large; this norm will be of order $r_0 L$, so for a large norm we need a large $r_0$; this will fail if $c^\ast$ vanishes for large $r_0$, but for the case of the Gross-Pitaevskii equation it will not. $\endgroup$
    – Carlo Beenakker
    Commented Feb 13, 2023 at 11:52
  • $\begingroup$ Thank you. I still have a question. In the case of Gross-Pitaevskii, If I am not mistaken, the $L^2$ norm of the profile $U$ cannot $<\!<1$, so there exists a lower bound>0 where the $L^2$ norm of the traveling waves of Gross-Pitaevskii cannot go below this bound. $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 13:02
  • $\begingroup$ yes, there is a lower bound but no upper bound. $\endgroup$
    – Carlo Beenakker
    Commented Feb 13, 2023 at 13:07
  • $\begingroup$ Sorry, I will not mark the answer as accepted, because my initial question (which probably I didn't formulate clearly) was to know if there exists a PDE admitting traveling waves without lower and upper bound in their $L^2$ norm... $\endgroup$
    – Niser
    Commented Feb 13, 2023 at 13:23
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I recently came across a paper on arXiv that addresses the question. I am sharing it here if it may be of interest to someone else.

The author seems to consider a nonlocal nonlinear schrödinger equation, referred to as the Calogero-Sutherland DNLS equation. And she finds periodic (in the space variable) traveling waves with small and large $L^2$-norms.

See Remark 1.3 (1) of page 5.

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